Statistical Inference Of Skew-normal Mixture For Joint Models

Normal distribution is one of the most common distributions in statistical analysis which requires the symmetry of response variables.However,there are asymmetric response variables in real life such as economic finance and environmental engineering and other fields.To study these problems,statisticians proposed many statistical asymmetric models and the most important of which is the skew-normal model.Most of the traditional skew-normal model are aim at modeling the mean parameters.But there are a lot of heteroscedastic data in practical application and mean regression model is difficult to fit heteroscedastic data exactly.Consequently,modeling a joint model is necessary which models the variance parameters at the same time.We also propose skewness parameter model in this paper to analyse the skewness parameters.In addition,in the era of big data,the complex data requires more statistical modeling.For heterogeneous population data,getting satisfactory results by using the traditional single model is difficult.That's the reason why mixture regression model is proposed.Mixture regression model is one of the important statistical analysis methods which studies heterogeneous population and is widely studied in statistics and machine learning.This paper mainly studies the following three aspects:Firstly,aiming at the skewness normal heterogeneous population data,the finite mixture of expert regression model with skew-normal data is established.Considering that the data may be heteroscedastic,the scale parameters are further modeled.MM and EM algorithms are used to study the maximum likelihood estimation of the mixture of expert regression models for joint location,scale with skew-normal data,and the specific steps of the algorithm are given.The results of the model and method are illustrated by simulation and analysis of real data of Kunming temperature.Secondly,on the basis of the first part,the mixture of expert regression models for joint location,scale and skewness with skew-normal data is proposed which models the skewness parameter.MM algorithm and EM algorithm are also used to estimate the parameters of the models and the effect of parameter estimation is illustrated by simulation analysis.Thirdly,based on the finite mixture of joint location and scale regression model,the problem of variable selection is studied with the skewness normal heterogeneous population data.According to three different penalty functions(SCAD,LASSO and HARD),the corresponding penalty likelihood function is obtained,and the BIC criterion is used to select the appropriate tuning parameters.The convergence rate of parameter estimation,the consistency of variable selection and the related asymptotic properties are proved.The simulation analysis and the example analysis of air quality index(AQI)data are given to illustrate the effectiveness of the proposed method.